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Linear Equations

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1. An art dealer sold two artworks for $1520 thereby making a profit of 25% on the first work and 10% profit on the other, whereas if he had approached any exhibition he would have sold them together for $1535 with a profit of 10% on the first and 25% on the other artwork. Find the actual cost of each artwork.

2. A professor started a project with a group of students, 60% of whom were boys. Due to some unavoidable reasons, six girls couldn't turn up, so the professor had to make some changes in the group. He admitted six boys. In doing so, the percentage of boys in the project increased to 75%. Find the number of boys and girls involved in the project initially.

3. Jane invested some amount at the rate of 12% simple interest and some other amount at the rate of 10% simple interest. She received yearly interest of $130. Randy also invested in the same scheme, but he interchanged the amounts invested, and received $4 more as interest. How much amount did each of them invest at different rates?

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Applications of Linear Equations

For all of these, the easiest method is to solve an equation for one variable, then plug the result into the second equation. The hardest part is figuring out what the equations are.

1. An art dealer sold two artworks for $1520 thereby making a profit of 25% on the first work and 10% profit on the other, whereas if he had approached any exhibition he would have sold them together for $1535 with a profit of 10% on the first and 25% on the other artwork. Find the actual cost of each artwork.

Let's let X be the cost to make the first piece of art and Y be the cost to make the second piece of art.

From the first sentence, we know that

1520 = 1.25X + 1.10Y

1520 is the total amount of money that he received for the two works, 1.25X is what he gets for the first artwork (X, the cost of making the art plus 0.25X, the profit), and 1.10Y is what he gets for the second artwork (Y, the cost of making the art plus 0.10Y, the profit).

By the same logic, we also have the ...

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